3 * This file contains the interface to the underlying bignum package.
4 * Its most important design principle is to completely hide the inner
5 * working of that other package from the user of GiNaC. It must either
6 * provide implementation of arithmetic operators and numerical evaluation
7 * of special functions or implement the interface to the bignum package. */
10 * GiNaC Copyright (C) 1999-2006 Johannes Gutenberg University Mainz, Germany
12 * This program is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU General Public License as published by
14 * the Free Software Foundation; either version 2 of the License, or
15 * (at your option) any later version.
17 * This program is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU General Public License for more details.
22 * You should have received a copy of the GNU General Public License
23 * along with this program; if not, write to the Free Software
24 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA
37 #include "operators.h"
42 // CLN should pollute the global namespace as little as possible. Hence, we
43 // include most of it here and include only the part needed for properly
44 // declaring cln::cl_number in numeric.h. This can only be safely done in
45 // namespaced versions of CLN, i.e. version > 1.1.0. Also, we only need a
46 // subset of CLN, so we don't include the complete <cln/cln.h> but only the
48 #include <cln/output.h>
49 #include <cln/integer_io.h>
50 #include <cln/integer_ring.h>
51 #include <cln/rational_io.h>
52 #include <cln/rational_ring.h>
53 #include <cln/lfloat_class.h>
54 #include <cln/lfloat_io.h>
55 #include <cln/real_io.h>
56 #include <cln/real_ring.h>
57 #include <cln/complex_io.h>
58 #include <cln/complex_ring.h>
59 #include <cln/numtheory.h>
63 GINAC_IMPLEMENT_REGISTERED_CLASS_OPT(numeric, basic,
64 print_func<print_context>(&numeric::do_print).
65 print_func<print_latex>(&numeric::do_print_latex).
66 print_func<print_csrc>(&numeric::do_print_csrc).
67 print_func<print_csrc_cl_N>(&numeric::do_print_csrc_cl_N).
68 print_func<print_tree>(&numeric::do_print_tree).
69 print_func<print_python_repr>(&numeric::do_print_python_repr))
72 // default constructor
75 /** default ctor. Numerically it initializes to an integer zero. */
76 numeric::numeric() : basic(&numeric::tinfo_static)
79 setflag(status_flags::evaluated | status_flags::expanded);
88 numeric::numeric(int i) : basic(&numeric::tinfo_static)
90 // Not the whole int-range is available if we don't cast to long
91 // first. This is due to the behaviour of the cl_I-ctor, which
92 // emphasizes efficiency. However, if the integer is small enough
93 // we save space and dereferences by using an immediate type.
94 // (C.f. <cln/object.h>)
95 // The #if clause prevents compiler warnings on 64bit machines where the
96 // comparision is always true.
97 #if cl_value_len >= 32
100 if (i < (1L << (cl_value_len-1)) && i >= -(1L << (cl_value_len-1)))
101 value = cln::cl_I(i);
103 value = cln::cl_I(static_cast<long>(i));
105 setflag(status_flags::evaluated | status_flags::expanded);
109 numeric::numeric(unsigned int i) : basic(&numeric::tinfo_static)
111 // Not the whole uint-range is available if we don't cast to ulong
112 // first. This is due to the behaviour of the cl_I-ctor, which
113 // emphasizes efficiency. However, if the integer is small enough
114 // we save space and dereferences by using an immediate type.
115 // (C.f. <cln/object.h>)
116 // The #if clause prevents compiler warnings on 64bit machines where the
117 // comparision is always true.
118 #if cl_value_len >= 32
119 value = cln::cl_I(i);
121 if (i < (1UL << (cl_value_len-1)))
122 value = cln::cl_I(i);
124 value = cln::cl_I(static_cast<unsigned long>(i));
126 setflag(status_flags::evaluated | status_flags::expanded);
130 numeric::numeric(long i) : basic(&numeric::tinfo_static)
132 value = cln::cl_I(i);
133 setflag(status_flags::evaluated | status_flags::expanded);
137 numeric::numeric(unsigned long i) : basic(&numeric::tinfo_static)
139 value = cln::cl_I(i);
140 setflag(status_flags::evaluated | status_flags::expanded);
144 /** Constructor for rational numerics a/b.
146 * @exception overflow_error (division by zero) */
147 numeric::numeric(long numer, long denom) : basic(&numeric::tinfo_static)
150 throw std::overflow_error("division by zero");
151 value = cln::cl_I(numer) / cln::cl_I(denom);
152 setflag(status_flags::evaluated | status_flags::expanded);
156 numeric::numeric(double d) : basic(&numeric::tinfo_static)
158 // We really want to explicitly use the type cl_LF instead of the
159 // more general cl_F, since that would give us a cl_DF only which
160 // will not be promoted to cl_LF if overflow occurs:
161 value = cln::cl_float(d, cln::default_float_format);
162 setflag(status_flags::evaluated | status_flags::expanded);
166 /** ctor from C-style string. It also accepts complex numbers in GiNaC
167 * notation like "2+5*I". */
168 numeric::numeric(const char *s) : basic(&numeric::tinfo_static)
170 cln::cl_N ctorval = 0;
171 // parse complex numbers (functional but not completely safe, unfortunately
172 // std::string does not understand regexpese):
173 // ss should represent a simple sum like 2+5*I
175 std::string::size_type delim;
177 // make this implementation safe by adding explicit sign
178 if (ss.at(0) != '+' && ss.at(0) != '-' && ss.at(0) != '#')
181 // We use 'E' as exponent marker in the output, but some people insist on
182 // writing 'e' at input, so let's substitute them right at the beginning:
183 while ((delim = ss.find("e"))!=std::string::npos)
184 ss.replace(delim,1,"E");
188 // chop ss into terms from left to right
190 bool imaginary = false;
191 delim = ss.find_first_of(std::string("+-"),1);
192 // Do we have an exponent marker like "31.415E-1"? If so, hop on!
193 if (delim!=std::string::npos && ss.at(delim-1)=='E')
194 delim = ss.find_first_of(std::string("+-"),delim+1);
195 term = ss.substr(0,delim);
196 if (delim!=std::string::npos)
197 ss = ss.substr(delim);
198 // is the term imaginary?
199 if (term.find("I")!=std::string::npos) {
201 term.erase(term.find("I"),1);
203 if (term.find("*")!=std::string::npos)
204 term.erase(term.find("*"),1);
205 // correct for trivial +/-I without explicit factor on I:
210 if (term.find('.')!=std::string::npos || term.find('E')!=std::string::npos) {
211 // CLN's short type cl_SF is not very useful within the GiNaC
212 // framework where we are mainly interested in the arbitrary
213 // precision type cl_LF. Hence we go straight to the construction
214 // of generic floats. In order to create them we have to convert
215 // our own floating point notation used for output and construction
216 // from char * to CLN's generic notation:
217 // 3.14 --> 3.14e0_<Digits>
218 // 31.4E-1 --> 31.4e-1_<Digits>
220 // No exponent marker? Let's add a trivial one.
221 if (term.find("E")==std::string::npos)
224 term = term.replace(term.find("E"),1,"e");
225 // append _<Digits> to term
226 term += "_" + ToString((unsigned)Digits);
227 // construct float using cln::cl_F(const char *) ctor.
229 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_F(term.c_str()));
231 ctorval = ctorval + cln::cl_F(term.c_str());
233 // this is not a floating point number...
235 ctorval = ctorval + cln::complex(cln::cl_I(0),cln::cl_R(term.c_str()));
237 ctorval = ctorval + cln::cl_R(term.c_str());
239 } while (delim != std::string::npos);
241 setflag(status_flags::evaluated | status_flags::expanded);
245 /** Ctor from CLN types. This is for the initiated user or internal use
247 numeric::numeric(const cln::cl_N &z) : basic(&numeric::tinfo_static)
250 setflag(status_flags::evaluated | status_flags::expanded);
258 numeric::numeric(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
260 cln::cl_N ctorval = 0;
262 // Read number as string
264 if (n.find_string("number", str)) {
265 std::istringstream s(str);
266 cln::cl_idecoded_float re, im;
270 case 'R': // Integer-decoded real number
271 s >> re.sign >> re.mantissa >> re.exponent;
272 ctorval = re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent);
274 case 'C': // Integer-decoded complex number
275 s >> re.sign >> re.mantissa >> re.exponent;
276 s >> im.sign >> im.mantissa >> im.exponent;
277 ctorval = cln::complex(re.sign * re.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), re.exponent),
278 im.sign * im.mantissa * cln::expt(cln::cl_float(2.0, cln::default_float_format), im.exponent));
280 default: // Ordinary number
287 setflag(status_flags::evaluated | status_flags::expanded);
290 void numeric::archive(archive_node &n) const
292 inherited::archive(n);
294 // Write number as string
295 std::ostringstream s;
296 if (this->is_crational())
299 // Non-rational numbers are written in an integer-decoded format
300 // to preserve the precision
301 if (this->is_real()) {
302 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(value));
304 s << re.sign << " " << re.mantissa << " " << re.exponent;
306 cln::cl_idecoded_float re = cln::integer_decode_float(cln::the<cln::cl_F>(cln::realpart(cln::the<cln::cl_N>(value))));
307 cln::cl_idecoded_float im = cln::integer_decode_float(cln::the<cln::cl_F>(cln::imagpart(cln::the<cln::cl_N>(value))));
309 s << re.sign << " " << re.mantissa << " " << re.exponent << " ";
310 s << im.sign << " " << im.mantissa << " " << im.exponent;
313 n.add_string("number", s.str());
316 DEFAULT_UNARCHIVE(numeric)
319 // functions overriding virtual functions from base classes
322 /** Helper function to print a real number in a nicer way than is CLN's
323 * default. Instead of printing 42.0L0 this just prints 42.0 to ostream os
324 * and instead of 3.99168L7 it prints 3.99168E7. This is fine in GiNaC as
325 * long as it only uses cl_LF and no other floating point types that we might
326 * want to visibly distinguish from cl_LF.
328 * @see numeric::print() */
329 static void print_real_number(const print_context & c, const cln::cl_R & x)
331 cln::cl_print_flags ourflags;
332 if (cln::instanceof(x, cln::cl_RA_ring)) {
333 // case 1: integer or rational
334 if (cln::instanceof(x, cln::cl_I_ring) ||
335 !is_a<print_latex>(c)) {
336 cln::print_real(c.s, ourflags, x);
337 } else { // rational output in LaTeX context
341 cln::print_real(c.s, ourflags, cln::abs(cln::numerator(cln::the<cln::cl_RA>(x))));
343 cln::print_real(c.s, ourflags, cln::denominator(cln::the<cln::cl_RA>(x)));
348 // make CLN believe this number has default_float_format, so it prints
349 // 'E' as exponent marker instead of 'L':
350 ourflags.default_float_format = cln::float_format(cln::the<cln::cl_F>(x));
351 cln::print_real(c.s, ourflags, x);
355 /** Helper function to print integer number in C++ source format.
357 * @see numeric::print() */
358 static void print_integer_csrc(const print_context & c, const cln::cl_I & x)
360 // Print small numbers in compact float format, but larger numbers in
362 const int max_cln_int = 536870911; // 2^29-1
363 if (x >= cln::cl_I(-max_cln_int) && x <= cln::cl_I(max_cln_int))
364 c.s << cln::cl_I_to_int(x) << ".0";
366 c.s << cln::double_approx(x);
369 /** Helper function to print real number in C++ source format.
371 * @see numeric::print() */
372 static void print_real_csrc(const print_context & c, const cln::cl_R & x)
374 if (cln::instanceof(x, cln::cl_I_ring)) {
377 print_integer_csrc(c, cln::the<cln::cl_I>(x));
379 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
382 const cln::cl_I numer = cln::numerator(cln::the<cln::cl_RA>(x));
383 const cln::cl_I denom = cln::denominator(cln::the<cln::cl_RA>(x));
384 if (cln::plusp(x) > 0) {
386 print_integer_csrc(c, numer);
389 print_integer_csrc(c, -numer);
392 print_integer_csrc(c, denom);
398 c.s << cln::double_approx(x);
402 /** Helper function to print real number in C++ source format using cl_N types.
404 * @see numeric::print() */
405 static void print_real_cl_N(const print_context & c, const cln::cl_R & x)
407 if (cln::instanceof(x, cln::cl_I_ring)) {
410 c.s << "cln::cl_I(\"";
411 print_real_number(c, x);
414 } else if (cln::instanceof(x, cln::cl_RA_ring)) {
417 cln::cl_print_flags ourflags;
418 c.s << "cln::cl_RA(\"";
419 cln::print_rational(c.s, ourflags, cln::the<cln::cl_RA>(x));
425 c.s << "cln::cl_F(\"";
426 print_real_number(c, cln::cl_float(1.0, cln::default_float_format) * x);
427 c.s << "_" << Digits << "\")";
431 void numeric::print_numeric(const print_context & c, const char *par_open, const char *par_close, const char *imag_sym, const char *mul_sym, unsigned level) const
433 const cln::cl_R r = cln::realpart(value);
434 const cln::cl_R i = cln::imagpart(value);
438 // case 1, real: x or -x
439 if ((precedence() <= level) && (!this->is_nonneg_integer())) {
441 print_real_number(c, r);
444 print_real_number(c, r);
450 // case 2, imaginary: y*I or -y*I
454 if (precedence()<=level)
457 c.s << "-" << imag_sym;
459 print_real_number(c, i);
460 c.s << mul_sym << imag_sym;
462 if (precedence()<=level)
468 // case 3, complex: x+y*I or x-y*I or -x+y*I or -x-y*I
469 if (precedence() <= level)
471 print_real_number(c, r);
474 c.s << "-" << imag_sym;
476 print_real_number(c, i);
477 c.s << mul_sym << imag_sym;
481 c.s << "+" << imag_sym;
484 print_real_number(c, i);
485 c.s << mul_sym << imag_sym;
488 if (precedence() <= level)
494 void numeric::do_print(const print_context & c, unsigned level) const
496 print_numeric(c, "(", ")", "I", "*", level);
499 void numeric::do_print_latex(const print_latex & c, unsigned level) const
501 print_numeric(c, "{(", ")}", "i", " ", level);
504 void numeric::do_print_csrc(const print_csrc & c, unsigned level) const
506 std::ios::fmtflags oldflags = c.s.flags();
507 c.s.setf(std::ios::scientific);
508 int oldprec = c.s.precision();
511 if (is_a<print_csrc_double>(c))
512 c.s.precision(std::numeric_limits<double>::digits10 + 1);
514 c.s.precision(std::numeric_limits<float>::digits10 + 1);
516 if (this->is_real()) {
519 print_real_csrc(c, cln::the<cln::cl_R>(value));
524 c.s << "std::complex<";
525 if (is_a<print_csrc_double>(c))
530 print_real_csrc(c, cln::realpart(value));
532 print_real_csrc(c, cln::imagpart(value));
537 c.s.precision(oldprec);
540 void numeric::do_print_csrc_cl_N(const print_csrc_cl_N & c, unsigned level) const
542 if (this->is_real()) {
545 print_real_cl_N(c, cln::the<cln::cl_R>(value));
550 c.s << "cln::complex(";
551 print_real_cl_N(c, cln::realpart(value));
553 print_real_cl_N(c, cln::imagpart(value));
558 void numeric::do_print_tree(const print_tree & c, unsigned level) const
560 c.s << std::string(level, ' ') << value
561 << " (" << class_name() << ")" << " @" << this
562 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
566 void numeric::do_print_python_repr(const print_python_repr & c, unsigned level) const
568 c.s << class_name() << "('";
569 print_numeric(c, "(", ")", "I", "*", level);
573 bool numeric::info(unsigned inf) const
576 case info_flags::numeric:
577 case info_flags::polynomial:
578 case info_flags::rational_function:
580 case info_flags::real:
582 case info_flags::rational:
583 case info_flags::rational_polynomial:
584 return is_rational();
585 case info_flags::crational:
586 case info_flags::crational_polynomial:
587 return is_crational();
588 case info_flags::integer:
589 case info_flags::integer_polynomial:
591 case info_flags::cinteger:
592 case info_flags::cinteger_polynomial:
593 return is_cinteger();
594 case info_flags::positive:
595 return is_positive();
596 case info_flags::negative:
597 return is_negative();
598 case info_flags::nonnegative:
599 return !is_negative();
600 case info_flags::posint:
601 return is_pos_integer();
602 case info_flags::negint:
603 return is_integer() && is_negative();
604 case info_flags::nonnegint:
605 return is_nonneg_integer();
606 case info_flags::even:
608 case info_flags::odd:
610 case info_flags::prime:
612 case info_flags::algebraic:
618 bool numeric::is_polynomial(const ex & var) const
623 int numeric::degree(const ex & s) const
628 int numeric::ldegree(const ex & s) const
633 ex numeric::coeff(const ex & s, int n) const
635 return n==0 ? *this : _ex0;
638 /** Disassemble real part and imaginary part to scan for the occurrence of a
639 * single number. Also handles the imaginary unit. It ignores the sign on
640 * both this and the argument, which may lead to what might appear as funny
641 * results: (2+I).has(-2) -> true. But this is consistent, since we also
642 * would like to have (-2+I).has(2) -> true and we want to think about the
643 * sign as a multiplicative factor. */
644 bool numeric::has(const ex &other, unsigned options) const
646 if (!is_exactly_a<numeric>(other))
648 const numeric &o = ex_to<numeric>(other);
649 if (this->is_equal(o) || this->is_equal(-o))
651 if (o.imag().is_zero()) { // e.g. scan for 3 in -3*I
652 if (!this->real().is_equal(*_num0_p))
653 if (this->real().is_equal(o) || this->real().is_equal(-o))
655 if (!this->imag().is_equal(*_num0_p))
656 if (this->imag().is_equal(o) || this->imag().is_equal(-o))
661 if (o.is_equal(I)) // e.g scan for I in 42*I
662 return !this->is_real();
663 if (o.real().is_zero()) // e.g. scan for 2*I in 2*I+1
664 if (!this->imag().is_equal(*_num0_p))
665 if (this->imag().is_equal(o*I) || this->imag().is_equal(-o*I))
672 /** Evaluation of numbers doesn't do anything at all. */
673 ex numeric::eval(int level) const
675 // Warning: if this is ever gonna do something, the ex ctors from all kinds
676 // of numbers should be checking for status_flags::evaluated.
681 /** Cast numeric into a floating-point object. For example exact numeric(1) is
682 * returned as a 1.0000000000000000000000 and so on according to how Digits is
683 * currently set. In case the object already was a floating point number the
684 * precision is trimmed to match the currently set default.
686 * @param level ignored, only needed for overriding basic::evalf.
687 * @return an ex-handle to a numeric. */
688 ex numeric::evalf(int level) const
690 // level can safely be discarded for numeric objects.
691 return numeric(cln::cl_float(1.0, cln::default_float_format) * value);
694 ex numeric::conjugate() const
699 return numeric(cln::conjugate(this->value));
702 ex numeric::real_part() const
704 return numeric(cln::realpart(value));
707 ex numeric::imag_part() const
709 return numeric(cln::imagpart(value));
714 int numeric::compare_same_type(const basic &other) const
716 GINAC_ASSERT(is_exactly_a<numeric>(other));
717 const numeric &o = static_cast<const numeric &>(other);
719 return this->compare(o);
723 bool numeric::is_equal_same_type(const basic &other) const
725 GINAC_ASSERT(is_exactly_a<numeric>(other));
726 const numeric &o = static_cast<const numeric &>(other);
728 return this->is_equal(o);
732 unsigned numeric::calchash() const
734 // Base computation of hashvalue on CLN's hashcode. Note: That depends
735 // only on the number's value, not its type or precision (i.e. a true
736 // equivalence relation on numbers). As a consequence, 3 and 3.0 share
737 // the same hashvalue. That shouldn't really matter, though.
738 setflag(status_flags::hash_calculated);
739 hashvalue = golden_ratio_hash(cln::equal_hashcode(value));
745 // new virtual functions which can be overridden by derived classes
751 // non-virtual functions in this class
756 /** Numerical addition method. Adds argument to *this and returns result as
757 * a numeric object. */
758 const numeric numeric::add(const numeric &other) const
760 return numeric(value + other.value);
764 /** Numerical subtraction method. Subtracts argument from *this and returns
765 * result as a numeric object. */
766 const numeric numeric::sub(const numeric &other) const
768 return numeric(value - other.value);
772 /** Numerical multiplication method. Multiplies *this and argument and returns
773 * result as a numeric object. */
774 const numeric numeric::mul(const numeric &other) const
776 return numeric(value * other.value);
780 /** Numerical division method. Divides *this by argument and returns result as
783 * @exception overflow_error (division by zero) */
784 const numeric numeric::div(const numeric &other) const
786 if (cln::zerop(other.value))
787 throw std::overflow_error("numeric::div(): division by zero");
788 return numeric(value / other.value);
792 /** Numerical exponentiation. Raises *this to the power given as argument and
793 * returns result as a numeric object. */
794 const numeric numeric::power(const numeric &other) const
796 // Shortcut for efficiency and numeric stability (as in 1.0 exponent):
797 // trap the neutral exponent.
798 if (&other==_num1_p || cln::equal(other.value,_num1_p->value))
801 if (cln::zerop(value)) {
802 if (cln::zerop(other.value))
803 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
804 else if (cln::zerop(cln::realpart(other.value)))
805 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
806 else if (cln::minusp(cln::realpart(other.value)))
807 throw std::overflow_error("numeric::eval(): division by zero");
811 return numeric(cln::expt(value, other.value));
816 /** Numerical addition method. Adds argument to *this and returns result as
817 * a numeric object on the heap. Use internally only for direct wrapping into
818 * an ex object, where the result would end up on the heap anyways. */
819 const numeric &numeric::add_dyn(const numeric &other) const
821 // Efficiency shortcut: trap the neutral element by pointer. This hack
822 // is supposed to keep the number of distinct numeric objects low.
825 else if (&other==_num0_p)
828 return static_cast<const numeric &>((new numeric(value + other.value))->
829 setflag(status_flags::dynallocated));
833 /** Numerical subtraction method. Subtracts argument from *this and returns
834 * result as a numeric object on the heap. Use internally only for direct
835 * wrapping into an ex object, where the result would end up on the heap
837 const numeric &numeric::sub_dyn(const numeric &other) const
839 // Efficiency shortcut: trap the neutral exponent (first by pointer). This
840 // hack is supposed to keep the number of distinct numeric objects low.
841 if (&other==_num0_p || cln::zerop(other.value))
844 return static_cast<const numeric &>((new numeric(value - other.value))->
845 setflag(status_flags::dynallocated));
849 /** Numerical multiplication method. Multiplies *this and argument and returns
850 * result as a numeric object on the heap. Use internally only for direct
851 * wrapping into an ex object, where the result would end up on the heap
853 const numeric &numeric::mul_dyn(const numeric &other) const
855 // Efficiency shortcut: trap the neutral element by pointer. This hack
856 // is supposed to keep the number of distinct numeric objects low.
859 else if (&other==_num1_p)
862 return static_cast<const numeric &>((new numeric(value * other.value))->
863 setflag(status_flags::dynallocated));
867 /** Numerical division method. Divides *this by argument and returns result as
868 * a numeric object on the heap. Use internally only for direct wrapping
869 * into an ex object, where the result would end up on the heap
872 * @exception overflow_error (division by zero) */
873 const numeric &numeric::div_dyn(const numeric &other) const
875 // Efficiency shortcut: trap the neutral element by pointer. This hack
876 // is supposed to keep the number of distinct numeric objects low.
879 if (cln::zerop(cln::the<cln::cl_N>(other.value)))
880 throw std::overflow_error("division by zero");
881 return static_cast<const numeric &>((new numeric(value / other.value))->
882 setflag(status_flags::dynallocated));
886 /** Numerical exponentiation. Raises *this to the power given as argument and
887 * returns result as a numeric object on the heap. Use internally only for
888 * direct wrapping into an ex object, where the result would end up on the
890 const numeric &numeric::power_dyn(const numeric &other) const
892 // Efficiency shortcut: trap the neutral exponent (first try by pointer, then
893 // try harder, since calls to cln::expt() below may return amazing results for
894 // floating point exponent 1.0).
895 if (&other==_num1_p || cln::equal(other.value, _num1_p->value))
898 if (cln::zerop(value)) {
899 if (cln::zerop(other.value))
900 throw std::domain_error("numeric::eval(): pow(0,0) is undefined");
901 else if (cln::zerop(cln::realpart(other.value)))
902 throw std::domain_error("numeric::eval(): pow(0,I) is undefined");
903 else if (cln::minusp(cln::realpart(other.value)))
904 throw std::overflow_error("numeric::eval(): division by zero");
908 return static_cast<const numeric &>((new numeric(cln::expt(value, other.value)))->
909 setflag(status_flags::dynallocated));
913 const numeric &numeric::operator=(int i)
915 return operator=(numeric(i));
919 const numeric &numeric::operator=(unsigned int i)
921 return operator=(numeric(i));
925 const numeric &numeric::operator=(long i)
927 return operator=(numeric(i));
931 const numeric &numeric::operator=(unsigned long i)
933 return operator=(numeric(i));
937 const numeric &numeric::operator=(double d)
939 return operator=(numeric(d));
943 const numeric &numeric::operator=(const char * s)
945 return operator=(numeric(s));
949 /** Inverse of a number. */
950 const numeric numeric::inverse() const
952 if (cln::zerop(value))
953 throw std::overflow_error("numeric::inverse(): division by zero");
954 return numeric(cln::recip(value));
957 /** Return the step function of a numeric. The imaginary part of it is
958 * ignored because the step function is generally considered real but
959 * a numeric may develop a small imaginary part due to rounding errors.
961 numeric numeric::step() const
962 { cln::cl_R r = cln::realpart(value);
970 /** Return the complex half-plane (left or right) in which the number lies.
971 * csgn(x)==0 for x==0, csgn(x)==1 for Re(x)>0 or Re(x)=0 and Im(x)>0,
972 * csgn(x)==-1 for Re(x)<0 or Re(x)=0 and Im(x)<0.
974 * @see numeric::compare(const numeric &other) */
975 int numeric::csgn() const
977 if (cln::zerop(value))
979 cln::cl_R r = cln::realpart(value);
980 if (!cln::zerop(r)) {
986 if (cln::plusp(cln::imagpart(value)))
994 /** This method establishes a canonical order on all numbers. For complex
995 * numbers this is not possible in a mathematically consistent way but we need
996 * to establish some order and it ought to be fast. So we simply define it
997 * to be compatible with our method csgn.
999 * @return csgn(*this-other)
1000 * @see numeric::csgn() */
1001 int numeric::compare(const numeric &other) const
1003 // Comparing two real numbers?
1004 if (cln::instanceof(value, cln::cl_R_ring) &&
1005 cln::instanceof(other.value, cln::cl_R_ring))
1006 // Yes, so just cln::compare them
1007 return cln::compare(cln::the<cln::cl_R>(value), cln::the<cln::cl_R>(other.value));
1009 // No, first cln::compare real parts...
1010 cl_signean real_cmp = cln::compare(cln::realpart(value), cln::realpart(other.value));
1013 // ...and then the imaginary parts.
1014 return cln::compare(cln::imagpart(value), cln::imagpart(other.value));
1019 bool numeric::is_equal(const numeric &other) const
1021 return cln::equal(value, other.value);
1025 /** True if object is zero. */
1026 bool numeric::is_zero() const
1028 return cln::zerop(value);
1032 /** True if object is not complex and greater than zero. */
1033 bool numeric::is_positive() const
1035 if (cln::instanceof(value, cln::cl_R_ring)) // real?
1036 return cln::plusp(cln::the<cln::cl_R>(value));
1041 /** True if object is not complex and less than zero. */
1042 bool numeric::is_negative() const
1044 if (cln::instanceof(value, cln::cl_R_ring)) // real?
1045 return cln::minusp(cln::the<cln::cl_R>(value));
1050 /** True if object is a non-complex integer. */
1051 bool numeric::is_integer() const
1053 return cln::instanceof(value, cln::cl_I_ring);
1057 /** True if object is an exact integer greater than zero. */
1058 bool numeric::is_pos_integer() const
1060 return (cln::instanceof(value, cln::cl_I_ring) && cln::plusp(cln::the<cln::cl_I>(value)));
1064 /** True if object is an exact integer greater or equal zero. */
1065 bool numeric::is_nonneg_integer() const
1067 return (cln::instanceof(value, cln::cl_I_ring) && !cln::minusp(cln::the<cln::cl_I>(value)));
1071 /** True if object is an exact even integer. */
1072 bool numeric::is_even() const
1074 return (cln::instanceof(value, cln::cl_I_ring) && cln::evenp(cln::the<cln::cl_I>(value)));
1078 /** True if object is an exact odd integer. */
1079 bool numeric::is_odd() const
1081 return (cln::instanceof(value, cln::cl_I_ring) && cln::oddp(cln::the<cln::cl_I>(value)));
1085 /** Probabilistic primality test.
1087 * @return true if object is exact integer and prime. */
1088 bool numeric::is_prime() const
1090 return (cln::instanceof(value, cln::cl_I_ring) // integer?
1091 && cln::plusp(cln::the<cln::cl_I>(value)) // positive?
1092 && cln::isprobprime(cln::the<cln::cl_I>(value)));
1096 /** True if object is an exact rational number, may even be complex
1097 * (denominator may be unity). */
1098 bool numeric::is_rational() const
1100 return cln::instanceof(value, cln::cl_RA_ring);
1104 /** True if object is a real integer, rational or float (but not complex). */
1105 bool numeric::is_real() const
1107 return cln::instanceof(value, cln::cl_R_ring);
1111 bool numeric::operator==(const numeric &other) const
1113 return cln::equal(value, other.value);
1117 bool numeric::operator!=(const numeric &other) const
1119 return !cln::equal(value, other.value);
1123 /** True if object is element of the domain of integers extended by I, i.e. is
1124 * of the form a+b*I, where a and b are integers. */
1125 bool numeric::is_cinteger() const
1127 if (cln::instanceof(value, cln::cl_I_ring))
1129 else if (!this->is_real()) { // complex case, handle n+m*I
1130 if (cln::instanceof(cln::realpart(value), cln::cl_I_ring) &&
1131 cln::instanceof(cln::imagpart(value), cln::cl_I_ring))
1138 /** True if object is an exact rational number, may even be complex
1139 * (denominator may be unity). */
1140 bool numeric::is_crational() const
1142 if (cln::instanceof(value, cln::cl_RA_ring))
1144 else if (!this->is_real()) { // complex case, handle Q(i):
1145 if (cln::instanceof(cln::realpart(value), cln::cl_RA_ring) &&
1146 cln::instanceof(cln::imagpart(value), cln::cl_RA_ring))
1153 /** Numerical comparison: less.
1155 * @exception invalid_argument (complex inequality) */
1156 bool numeric::operator<(const numeric &other) const
1158 if (this->is_real() && other.is_real())
1159 return (cln::the<cln::cl_R>(value) < cln::the<cln::cl_R>(other.value));
1160 throw std::invalid_argument("numeric::operator<(): complex inequality");
1164 /** Numerical comparison: less or equal.
1166 * @exception invalid_argument (complex inequality) */
1167 bool numeric::operator<=(const numeric &other) const
1169 if (this->is_real() && other.is_real())
1170 return (cln::the<cln::cl_R>(value) <= cln::the<cln::cl_R>(other.value));
1171 throw std::invalid_argument("numeric::operator<=(): complex inequality");
1175 /** Numerical comparison: greater.
1177 * @exception invalid_argument (complex inequality) */
1178 bool numeric::operator>(const numeric &other) const
1180 if (this->is_real() && other.is_real())
1181 return (cln::the<cln::cl_R>(value) > cln::the<cln::cl_R>(other.value));
1182 throw std::invalid_argument("numeric::operator>(): complex inequality");
1186 /** Numerical comparison: greater or equal.
1188 * @exception invalid_argument (complex inequality) */
1189 bool numeric::operator>=(const numeric &other) const
1191 if (this->is_real() && other.is_real())
1192 return (cln::the<cln::cl_R>(value) >= cln::the<cln::cl_R>(other.value));
1193 throw std::invalid_argument("numeric::operator>=(): complex inequality");
1197 /** Converts numeric types to machine's int. You should check with
1198 * is_integer() if the number is really an integer before calling this method.
1199 * You may also consider checking the range first. */
1200 int numeric::to_int() const
1202 GINAC_ASSERT(this->is_integer());
1203 return cln::cl_I_to_int(cln::the<cln::cl_I>(value));
1207 /** Converts numeric types to machine's long. You should check with
1208 * is_integer() if the number is really an integer before calling this method.
1209 * You may also consider checking the range first. */
1210 long numeric::to_long() const
1212 GINAC_ASSERT(this->is_integer());
1213 return cln::cl_I_to_long(cln::the<cln::cl_I>(value));
1217 /** Converts numeric types to machine's double. You should check with is_real()
1218 * if the number is really not complex before calling this method. */
1219 double numeric::to_double() const
1221 GINAC_ASSERT(this->is_real());
1222 return cln::double_approx(cln::realpart(value));
1226 /** Returns a new CLN object of type cl_N, representing the value of *this.
1227 * This method may be used when mixing GiNaC and CLN in one project.
1229 cln::cl_N numeric::to_cl_N() const
1235 /** Real part of a number. */
1236 const numeric numeric::real() const
1238 return numeric(cln::realpart(value));
1242 /** Imaginary part of a number. */
1243 const numeric numeric::imag() const
1245 return numeric(cln::imagpart(value));
1249 /** Numerator. Computes the numerator of rational numbers, rationalized
1250 * numerator of complex if real and imaginary part are both rational numbers
1251 * (i.e numer(4/3+5/6*I) == 8+5*I), the number carrying the sign in all other
1253 const numeric numeric::numer() const
1255 if (cln::instanceof(value, cln::cl_I_ring))
1256 return numeric(*this); // integer case
1258 else if (cln::instanceof(value, cln::cl_RA_ring))
1259 return numeric(cln::numerator(cln::the<cln::cl_RA>(value)));
1261 else if (!this->is_real()) { // complex case, handle Q(i):
1262 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
1263 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
1264 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1265 return numeric(*this);
1266 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1267 return numeric(cln::complex(r*cln::denominator(i), cln::numerator(i)));
1268 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1269 return numeric(cln::complex(cln::numerator(r), i*cln::denominator(r)));
1270 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring)) {
1271 const cln::cl_I s = cln::lcm(cln::denominator(r), cln::denominator(i));
1272 return numeric(cln::complex(cln::numerator(r)*(cln::exquo(s,cln::denominator(r))),
1273 cln::numerator(i)*(cln::exquo(s,cln::denominator(i)))));
1276 // at least one float encountered
1277 return numeric(*this);
1281 /** Denominator. Computes the denominator of rational numbers, common integer
1282 * denominator of complex if real and imaginary part are both rational numbers
1283 * (i.e denom(4/3+5/6*I) == 6), one in all other cases. */
1284 const numeric numeric::denom() const
1286 if (cln::instanceof(value, cln::cl_I_ring))
1287 return *_num1_p; // integer case
1289 if (cln::instanceof(value, cln::cl_RA_ring))
1290 return numeric(cln::denominator(cln::the<cln::cl_RA>(value)));
1292 if (!this->is_real()) { // complex case, handle Q(i):
1293 const cln::cl_RA r = cln::the<cln::cl_RA>(cln::realpart(value));
1294 const cln::cl_RA i = cln::the<cln::cl_RA>(cln::imagpart(value));
1295 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_I_ring))
1297 if (cln::instanceof(r, cln::cl_I_ring) && cln::instanceof(i, cln::cl_RA_ring))
1298 return numeric(cln::denominator(i));
1299 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_I_ring))
1300 return numeric(cln::denominator(r));
1301 if (cln::instanceof(r, cln::cl_RA_ring) && cln::instanceof(i, cln::cl_RA_ring))
1302 return numeric(cln::lcm(cln::denominator(r), cln::denominator(i)));
1304 // at least one float encountered
1309 /** Size in binary notation. For integers, this is the smallest n >= 0 such
1310 * that -2^n <= x < 2^n. If x > 0, this is the unique n > 0 such that
1311 * 2^(n-1) <= x < 2^n.
1313 * @return number of bits (excluding sign) needed to represent that number
1314 * in two's complement if it is an integer, 0 otherwise. */
1315 int numeric::int_length() const
1317 if (cln::instanceof(value, cln::cl_I_ring))
1318 return cln::integer_length(cln::the<cln::cl_I>(value));
1327 /** Imaginary unit. This is not a constant but a numeric since we are
1328 * natively handing complex numbers anyways, so in each expression containing
1329 * an I it is automatically eval'ed away anyhow. */
1330 const numeric I = numeric(cln::complex(cln::cl_I(0),cln::cl_I(1)));
1333 /** Exponential function.
1335 * @return arbitrary precision numerical exp(x). */
1336 const numeric exp(const numeric &x)
1338 return cln::exp(x.to_cl_N());
1342 /** Natural logarithm.
1344 * @param x complex number
1345 * @return arbitrary precision numerical log(x).
1346 * @exception pole_error("log(): logarithmic pole",0) */
1347 const numeric log(const numeric &x)
1350 throw pole_error("log(): logarithmic pole",0);
1351 return cln::log(x.to_cl_N());
1355 /** Numeric sine (trigonometric function).
1357 * @return arbitrary precision numerical sin(x). */
1358 const numeric sin(const numeric &x)
1360 return cln::sin(x.to_cl_N());
1364 /** Numeric cosine (trigonometric function).
1366 * @return arbitrary precision numerical cos(x). */
1367 const numeric cos(const numeric &x)
1369 return cln::cos(x.to_cl_N());
1373 /** Numeric tangent (trigonometric function).
1375 * @return arbitrary precision numerical tan(x). */
1376 const numeric tan(const numeric &x)
1378 return cln::tan(x.to_cl_N());
1382 /** Numeric inverse sine (trigonometric function).
1384 * @return arbitrary precision numerical asin(x). */
1385 const numeric asin(const numeric &x)
1387 return cln::asin(x.to_cl_N());
1391 /** Numeric inverse cosine (trigonometric function).
1393 * @return arbitrary precision numerical acos(x). */
1394 const numeric acos(const numeric &x)
1396 return cln::acos(x.to_cl_N());
1402 * @param x complex number
1404 * @exception pole_error("atan(): logarithmic pole",0) */
1405 const numeric atan(const numeric &x)
1408 x.real().is_zero() &&
1409 abs(x.imag()).is_equal(*_num1_p))
1410 throw pole_error("atan(): logarithmic pole",0);
1411 return cln::atan(x.to_cl_N());
1417 * @param x real number
1418 * @param y real number
1419 * @return atan(y/x) */
1420 const numeric atan(const numeric &y, const numeric &x)
1422 if (x.is_real() && y.is_real())
1423 return cln::atan(cln::the<cln::cl_R>(x.to_cl_N()),
1424 cln::the<cln::cl_R>(y.to_cl_N()));
1426 throw std::invalid_argument("atan(): complex argument");
1430 /** Numeric hyperbolic sine (trigonometric function).
1432 * @return arbitrary precision numerical sinh(x). */
1433 const numeric sinh(const numeric &x)
1435 return cln::sinh(x.to_cl_N());
1439 /** Numeric hyperbolic cosine (trigonometric function).
1441 * @return arbitrary precision numerical cosh(x). */
1442 const numeric cosh(const numeric &x)
1444 return cln::cosh(x.to_cl_N());
1448 /** Numeric hyperbolic tangent (trigonometric function).
1450 * @return arbitrary precision numerical tanh(x). */
1451 const numeric tanh(const numeric &x)
1453 return cln::tanh(x.to_cl_N());
1457 /** Numeric inverse hyperbolic sine (trigonometric function).
1459 * @return arbitrary precision numerical asinh(x). */
1460 const numeric asinh(const numeric &x)
1462 return cln::asinh(x.to_cl_N());
1466 /** Numeric inverse hyperbolic cosine (trigonometric function).
1468 * @return arbitrary precision numerical acosh(x). */
1469 const numeric acosh(const numeric &x)
1471 return cln::acosh(x.to_cl_N());
1475 /** Numeric inverse hyperbolic tangent (trigonometric function).
1477 * @return arbitrary precision numerical atanh(x). */
1478 const numeric atanh(const numeric &x)
1480 return cln::atanh(x.to_cl_N());
1484 /*static cln::cl_N Li2_series(const ::cl_N &x,
1485 const ::float_format_t &prec)
1487 // Note: argument must be in the unit circle
1488 // This is very inefficient unless we have fast floating point Bernoulli
1489 // numbers implemented!
1490 cln::cl_N c1 = -cln::log(1-x);
1492 // hard-wire the first two Bernoulli numbers
1493 cln::cl_N acc = c1 - cln::square(c1)/4;
1495 cln::cl_F pisq = cln::square(cln::cl_pi(prec)); // pi^2
1496 cln::cl_F piac = cln::cl_float(1, prec); // accumulator: pi^(2*i)
1498 c1 = cln::square(c1);
1502 aug = c2 * (*(bernoulli(numeric(2*i)).clnptr())) / cln::factorial(2*i+1);
1503 // aug = c2 * cln::cl_I(i%2 ? 1 : -1) / cln::cl_I(2*i+1) * cln::cl_zeta(2*i, prec) / piac / (cln::cl_I(1)<<(2*i-1));
1506 } while (acc != acc+aug);
1510 /** Numeric evaluation of Dilogarithm within circle of convergence (unit
1511 * circle) using a power series. */
1512 static cln::cl_N Li2_series(const cln::cl_N &x,
1513 const cln::float_format_t &prec)
1515 // Note: argument must be in the unit circle
1517 cln::cl_N num = cln::complex(cln::cl_float(1, prec), 0);
1522 den = den + i; // 1, 4, 9, 16, ...
1526 } while (acc != acc+aug);
1530 /** Folds Li2's argument inside a small rectangle to enhance convergence. */
1531 static cln::cl_N Li2_projection(const cln::cl_N &x,
1532 const cln::float_format_t &prec)
1534 const cln::cl_R re = cln::realpart(x);
1535 const cln::cl_R im = cln::imagpart(x);
1536 if (re > cln::cl_F(".5"))
1537 // zeta(2) - Li2(1-x) - log(x)*log(1-x)
1539 - Li2_series(1-x, prec)
1540 - cln::log(x)*cln::log(1-x));
1541 if ((re <= 0 && cln::abs(im) > cln::cl_F(".75")) || (re < cln::cl_F("-.5")))
1542 // -log(1-x)^2 / 2 - Li2(x/(x-1))
1543 return(- cln::square(cln::log(1-x))/2
1544 - Li2_series(x/(x-1), prec));
1545 if (re > 0 && cln::abs(im) > cln::cl_LF(".75"))
1546 // Li2(x^2)/2 - Li2(-x)
1547 return(Li2_projection(cln::square(x), prec)/2
1548 - Li2_projection(-x, prec));
1549 return Li2_series(x, prec);
1552 /** Numeric evaluation of Dilogarithm. The domain is the entire complex plane,
1553 * the branch cut lies along the positive real axis, starting at 1 and
1554 * continuous with quadrant IV.
1556 * @return arbitrary precision numerical Li2(x). */
1557 const numeric Li2(const numeric &x)
1562 // what is the desired float format?
1563 // first guess: default format
1564 cln::float_format_t prec = cln::default_float_format;
1565 const cln::cl_N value = x.to_cl_N();
1566 // second guess: the argument's format
1567 if (!x.real().is_rational())
1568 prec = cln::float_format(cln::the<cln::cl_F>(cln::realpart(value)));
1569 else if (!x.imag().is_rational())
1570 prec = cln::float_format(cln::the<cln::cl_F>(cln::imagpart(value)));
1572 if (value==1) // may cause trouble with log(1-x)
1573 return cln::zeta(2, prec);
1575 if (cln::abs(value) > 1)
1576 // -log(-x)^2 / 2 - zeta(2) - Li2(1/x)
1577 return(- cln::square(cln::log(-value))/2
1578 - cln::zeta(2, prec)
1579 - Li2_projection(cln::recip(value), prec));
1581 return Li2_projection(x.to_cl_N(), prec);
1585 /** Numeric evaluation of Riemann's Zeta function. Currently works only for
1586 * integer arguments. */
1587 const numeric zeta(const numeric &x)
1589 // A dirty hack to allow for things like zeta(3.0), since CLN currently
1590 // only knows about integer arguments and zeta(3).evalf() automatically
1591 // cascades down to zeta(3.0).evalf(). The trick is to rely on 3.0-3
1592 // being an exact zero for CLN, which can be tested and then we can just
1593 // pass the number casted to an int:
1595 const int aux = (int)(cln::double_approx(cln::the<cln::cl_R>(x.to_cl_N())));
1596 if (cln::zerop(x.to_cl_N()-aux))
1597 return cln::zeta(aux);
1603 /** The Gamma function.
1604 * This is only a stub! */
1605 const numeric lgamma(const numeric &x)
1609 const numeric tgamma(const numeric &x)
1615 /** The psi function (aka polygamma function).
1616 * This is only a stub! */
1617 const numeric psi(const numeric &x)
1623 /** The psi functions (aka polygamma functions).
1624 * This is only a stub! */
1625 const numeric psi(const numeric &n, const numeric &x)
1631 /** Factorial combinatorial function.
1633 * @param n integer argument >= 0
1634 * @exception range_error (argument must be integer >= 0) */
1635 const numeric factorial(const numeric &n)
1637 if (!n.is_nonneg_integer())
1638 throw std::range_error("numeric::factorial(): argument must be integer >= 0");
1639 return numeric(cln::factorial(n.to_int()));
1643 /** The double factorial combinatorial function. (Scarcely used, but still
1644 * useful in cases, like for exact results of tgamma(n+1/2) for instance.)
1646 * @param n integer argument >= -1
1647 * @return n!! == n * (n-2) * (n-4) * ... * ({1|2}) with 0!! == (-1)!! == 1
1648 * @exception range_error (argument must be integer >= -1) */
1649 const numeric doublefactorial(const numeric &n)
1651 if (n.is_equal(*_num_1_p))
1654 if (!n.is_nonneg_integer())
1655 throw std::range_error("numeric::doublefactorial(): argument must be integer >= -1");
1657 return numeric(cln::doublefactorial(n.to_int()));
1661 /** The Binomial coefficients. It computes the binomial coefficients. For
1662 * integer n and k and positive n this is the number of ways of choosing k
1663 * objects from n distinct objects. If n is negative, the formula
1664 * binomial(n,k) == (-1)^k*binomial(k-n-1,k) is used to compute the result. */
1665 const numeric binomial(const numeric &n, const numeric &k)
1667 if (n.is_integer() && k.is_integer()) {
1668 if (n.is_nonneg_integer()) {
1669 if (k.compare(n)!=1 && k.compare(*_num0_p)!=-1)
1670 return numeric(cln::binomial(n.to_int(),k.to_int()));
1674 return _num_1_p->power(k)*binomial(k-n-(*_num1_p),k);
1678 // should really be gamma(n+1)/gamma(k+1)/gamma(n-k+1) or a suitable limit
1679 throw std::range_error("numeric::binomial(): don't know how to evaluate that.");
1683 /** Bernoulli number. The nth Bernoulli number is the coefficient of x^n/n!
1684 * in the expansion of the function x/(e^x-1).
1686 * @return the nth Bernoulli number (a rational number).
1687 * @exception range_error (argument must be integer >= 0) */
1688 const numeric bernoulli(const numeric &nn)
1690 if (!nn.is_integer() || nn.is_negative())
1691 throw std::range_error("numeric::bernoulli(): argument must be integer >= 0");
1695 // The Bernoulli numbers are rational numbers that may be computed using
1698 // B_n = - 1/(n+1) * sum_{k=0}^{n-1}(binomial(n+1,k)*B_k)
1700 // with B(0) = 1. Since the n'th Bernoulli number depends on all the
1701 // previous ones, the computation is necessarily very expensive. There are
1702 // several other ways of computing them, a particularly good one being
1706 // for (unsigned i=0; i<n; i++) {
1707 // c = exquo(c*(i-n),(i+2));
1708 // Bern = Bern + c*s/(i+2);
1709 // s = s + expt_pos(cl_I(i+2),n);
1713 // But if somebody works with the n'th Bernoulli number she is likely to
1714 // also need all previous Bernoulli numbers. So we need a complete remember
1715 // table and above divide and conquer algorithm is not suited to build one
1716 // up. The formula below accomplishes this. It is a modification of the
1717 // defining formula above but the computation of the binomial coefficients
1718 // is carried along in an inline fashion. It also honors the fact that
1719 // B_n is zero when n is odd and greater than 1.
1721 // (There is an interesting relation with the tangent polynomials described
1722 // in `Concrete Mathematics', which leads to a program a little faster as
1723 // our implementation below, but it requires storing one such polynomial in
1724 // addition to the remember table. This doubles the memory footprint so
1725 // we don't use it.)
1727 const unsigned n = nn.to_int();
1729 // the special cases not covered by the algorithm below
1731 return (n==1) ? (*_num_1_2_p) : (*_num0_p);
1735 // store nonvanishing Bernoulli numbers here
1736 static std::vector< cln::cl_RA > results;
1737 static unsigned next_r = 0;
1739 // algorithm not applicable to B(2), so just store it
1741 results.push_back(cln::recip(cln::cl_RA(6)));
1745 return results[n/2-1];
1747 results.reserve(n/2);
1748 for (unsigned p=next_r; p<=n; p+=2) {
1749 cln::cl_I c = 1; // seed for binonmial coefficients
1750 cln::cl_RA b = cln::cl_RA(p-1)/-2;
1751 // The CLN manual says: "The conversion from `unsigned int' works only
1752 // if the argument is < 2^29" (This is for 32 Bit machines. More
1753 // generally, cl_value_len is the limiting exponent of 2. We must make
1754 // sure that no intermediates are created which exceed this value. The
1755 // largest intermediate is (p+3-2*k)*(p/2-k+1) <= (p^2+p)/2.
1756 if (p < (1UL<<cl_value_len/2)) {
1757 for (unsigned k=1; k<=p/2-1; ++k) {
1758 c = cln::exquo(c * ((p+3-2*k) * (p/2-k+1)), (2*k-1)*k);
1759 b = b + c*results[k-1];
1762 for (unsigned k=1; k<=p/2-1; ++k) {
1763 c = cln::exquo((c * (p+3-2*k)) * (p/2-k+1), cln::cl_I(2*k-1)*k);
1764 b = b + c*results[k-1];
1767 results.push_back(-b/(p+1));
1770 return results[n/2-1];
1774 /** Fibonacci number. The nth Fibonacci number F(n) is defined by the
1775 * recurrence formula F(n)==F(n-1)+F(n-2) with F(0)==0 and F(1)==1.
1777 * @param n an integer
1778 * @return the nth Fibonacci number F(n) (an integer number)
1779 * @exception range_error (argument must be an integer) */
1780 const numeric fibonacci(const numeric &n)
1782 if (!n.is_integer())
1783 throw std::range_error("numeric::fibonacci(): argument must be integer");
1786 // The following addition formula holds:
1788 // F(n+m) = F(m-1)*F(n) + F(m)*F(n+1) for m >= 1, n >= 0.
1790 // (Proof: For fixed m, the LHS and the RHS satisfy the same recurrence
1791 // w.r.t. n, and the initial values (n=0, n=1) agree. Hence all values
1793 // Replace m by m+1:
1794 // F(n+m+1) = F(m)*F(n) + F(m+1)*F(n+1) for m >= 0, n >= 0
1795 // Now put in m = n, to get
1796 // F(2n) = (F(n+1)-F(n))*F(n) + F(n)*F(n+1) = F(n)*(2*F(n+1) - F(n))
1797 // F(2n+1) = F(n)^2 + F(n+1)^2
1799 // F(2n+2) = F(n+1)*(2*F(n) + F(n+1))
1802 if (n.is_negative())
1804 return -fibonacci(-n);
1806 return fibonacci(-n);
1810 cln::cl_I m = cln::the<cln::cl_I>(n.to_cl_N()) >> 1L; // floor(n/2);
1811 for (uintL bit=cln::integer_length(m); bit>0; --bit) {
1812 // Since a squaring is cheaper than a multiplication, better use
1813 // three squarings instead of one multiplication and two squarings.
1814 cln::cl_I u2 = cln::square(u);
1815 cln::cl_I v2 = cln::square(v);
1816 if (cln::logbitp(bit-1, m)) {
1817 v = cln::square(u + v) - u2;
1820 u = v2 - cln::square(v - u);
1825 // Here we don't use the squaring formula because one multiplication
1826 // is cheaper than two squarings.
1827 return u * ((v << 1) - u);
1829 return cln::square(u) + cln::square(v);
1833 /** Absolute value. */
1834 const numeric abs(const numeric& x)
1836 return cln::abs(x.to_cl_N());
1840 /** Modulus (in positive representation).
1841 * In general, mod(a,b) has the sign of b or is zero, and rem(a,b) has the
1842 * sign of a or is zero. This is different from Maple's modp, where the sign
1843 * of b is ignored. It is in agreement with Mathematica's Mod.
1845 * @return a mod b in the range [0,abs(b)-1] with sign of b if both are
1846 * integer, 0 otherwise. */
1847 const numeric mod(const numeric &a, const numeric &b)
1849 if (a.is_integer() && b.is_integer())
1850 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()),
1851 cln::the<cln::cl_I>(b.to_cl_N()));
1857 /** Modulus (in symmetric representation).
1858 * Equivalent to Maple's mods.
1860 * @return a mod b in the range [-iquo(abs(b)-1,2), iquo(abs(b),2)]. */
1861 const numeric smod(const numeric &a, const numeric &b)
1863 if (a.is_integer() && b.is_integer()) {
1864 const cln::cl_I b2 = cln::ceiling1(cln::the<cln::cl_I>(b.to_cl_N()) >> 1) - 1;
1865 return cln::mod(cln::the<cln::cl_I>(a.to_cl_N()) + b2,
1866 cln::the<cln::cl_I>(b.to_cl_N())) - b2;
1872 /** Numeric integer remainder.
1873 * Equivalent to Maple's irem(a,b) as far as sign conventions are concerned.
1874 * In general, mod(a,b) has the sign of b or is zero, and irem(a,b) has the
1875 * sign of a or is zero.
1877 * @return remainder of a/b if both are integer, 0 otherwise.
1878 * @exception overflow_error (division by zero) if b is zero. */
1879 const numeric irem(const numeric &a, const numeric &b)
1882 throw std::overflow_error("numeric::irem(): division by zero");
1883 if (a.is_integer() && b.is_integer())
1884 return cln::rem(cln::the<cln::cl_I>(a.to_cl_N()),
1885 cln::the<cln::cl_I>(b.to_cl_N()));
1891 /** Numeric integer remainder.
1892 * Equivalent to Maple's irem(a,b,'q') it obeyes the relation
1893 * irem(a,b,q) == a - q*b. In general, mod(a,b) has the sign of b or is zero,
1894 * and irem(a,b) has the sign of a or is zero.
1896 * @return remainder of a/b and quotient stored in q if both are integer,
1898 * @exception overflow_error (division by zero) if b is zero. */
1899 const numeric irem(const numeric &a, const numeric &b, numeric &q)
1902 throw std::overflow_error("numeric::irem(): division by zero");
1903 if (a.is_integer() && b.is_integer()) {
1904 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1905 cln::the<cln::cl_I>(b.to_cl_N()));
1906 q = rem_quo.quotient;
1907 return rem_quo.remainder;
1915 /** Numeric integer quotient.
1916 * Equivalent to Maple's iquo as far as sign conventions are concerned.
1918 * @return truncated quotient of a/b if both are integer, 0 otherwise.
1919 * @exception overflow_error (division by zero) if b is zero. */
1920 const numeric iquo(const numeric &a, const numeric &b)
1923 throw std::overflow_error("numeric::iquo(): division by zero");
1924 if (a.is_integer() && b.is_integer())
1925 return cln::truncate1(cln::the<cln::cl_I>(a.to_cl_N()),
1926 cln::the<cln::cl_I>(b.to_cl_N()));
1932 /** Numeric integer quotient.
1933 * Equivalent to Maple's iquo(a,b,'r') it obeyes the relation
1934 * r == a - iquo(a,b,r)*b.
1936 * @return truncated quotient of a/b and remainder stored in r if both are
1937 * integer, 0 otherwise.
1938 * @exception overflow_error (division by zero) if b is zero. */
1939 const numeric iquo(const numeric &a, const numeric &b, numeric &r)
1942 throw std::overflow_error("numeric::iquo(): division by zero");
1943 if (a.is_integer() && b.is_integer()) {
1944 const cln::cl_I_div_t rem_quo = cln::truncate2(cln::the<cln::cl_I>(a.to_cl_N()),
1945 cln::the<cln::cl_I>(b.to_cl_N()));
1946 r = rem_quo.remainder;
1947 return rem_quo.quotient;
1955 /** Greatest Common Divisor.
1957 * @return The GCD of two numbers if both are integer, a numerical 1
1958 * if they are not. */
1959 const numeric gcd(const numeric &a, const numeric &b)
1961 if (a.is_integer() && b.is_integer())
1962 return cln::gcd(cln::the<cln::cl_I>(a.to_cl_N()),
1963 cln::the<cln::cl_I>(b.to_cl_N()));
1969 /** Least Common Multiple.
1971 * @return The LCM of two numbers if both are integer, the product of those
1972 * two numbers if they are not. */
1973 const numeric lcm(const numeric &a, const numeric &b)
1975 if (a.is_integer() && b.is_integer())
1976 return cln::lcm(cln::the<cln::cl_I>(a.to_cl_N()),
1977 cln::the<cln::cl_I>(b.to_cl_N()));
1983 /** Numeric square root.
1984 * If possible, sqrt(x) should respect squares of exact numbers, i.e. sqrt(4)
1985 * should return integer 2.
1987 * @param x numeric argument
1988 * @return square root of x. Branch cut along negative real axis, the negative
1989 * real axis itself where imag(x)==0 and real(x)<0 belongs to the upper part
1990 * where imag(x)>0. */
1991 const numeric sqrt(const numeric &x)
1993 return cln::sqrt(x.to_cl_N());
1997 /** Integer numeric square root. */
1998 const numeric isqrt(const numeric &x)
2000 if (x.is_integer()) {
2002 cln::isqrt(cln::the<cln::cl_I>(x.to_cl_N()), &root);
2009 /** Floating point evaluation of Archimedes' constant Pi. */
2012 return numeric(cln::pi(cln::default_float_format));
2016 /** Floating point evaluation of Euler's constant gamma. */
2019 return numeric(cln::eulerconst(cln::default_float_format));
2023 /** Floating point evaluation of Catalan's constant. */
2026 return numeric(cln::catalanconst(cln::default_float_format));
2030 /** _numeric_digits default ctor, checking for singleton invariance. */
2031 _numeric_digits::_numeric_digits()
2034 // It initializes to 17 digits, because in CLN float_format(17) turns out
2035 // to be 61 (<64) while float_format(18)=65. The reason is we want to
2036 // have a cl_LF instead of cl_SF, cl_FF or cl_DF.
2038 throw(std::runtime_error("I told you not to do instantiate me!"));
2040 cln::default_float_format = cln::float_format(17);
2042 // add callbacks for built-in functions
2043 // like ... add_callback(Li_lookuptable);
2047 /** Assign a native long to global Digits object. */
2048 _numeric_digits& _numeric_digits::operator=(long prec)
2050 long digitsdiff = prec - digits;
2052 cln::default_float_format = cln::float_format(prec);
2054 // call registered callbacks
2055 std::vector<digits_changed_callback>::const_iterator it = callbacklist.begin(), end = callbacklist.end();
2056 for (; it != end; ++it) {
2064 /** Convert global Digits object to native type long. */
2065 _numeric_digits::operator long()
2067 // BTW, this is approx. unsigned(cln::default_float_format*0.301)-1
2068 return (long)digits;
2072 /** Append global Digits object to ostream. */
2073 void _numeric_digits::print(std::ostream &os) const
2079 /** Add a new callback function. */
2080 void _numeric_digits::add_callback(digits_changed_callback callback)
2082 callbacklist.push_back(callback);
2086 std::ostream& operator<<(std::ostream &os, const _numeric_digits &e)
2093 // static member variables
2098 bool _numeric_digits::too_late = false;
2101 /** Accuracy in decimal digits. Only object of this type! Can be set using
2102 * assignment from C++ unsigned ints and evaluated like any built-in type. */
2103 _numeric_digits Digits;
2105 } // namespace GiNaC